This is the second part of a two-part paper considering bidirectional relay networks with half-duplex nodes and block fading where the nodes transmit with a fixed transmission rate. In Part I, it was shown that a considerable gain in terms of sum throughput can be obtained by optimally selecting the transmission modes or, equivalently, the states of the nodes, i.e., the transmit, the receive, and the silent states, based on the qualities of the involved links. To enable adaptive transmission mode selection, the relay has to be equipped with two buffers for storage of the data received from the two users. The protocol proposed in Part I was delay unconstrained and provides an upper bound for the performance of practical delay-constrained protocols. In this paper, we propose two heuristic but efficient delay-constrained protocols, which can approach the performance upper bound reported in Part I, even in cases where only a small delay is permitted. The proposed protocols not only consider the instantaneous qualities of the involved links for adaptive mode selection but also take the states of the queues at the buffers into account, i.e., the number of packets in the queues. The average throughput and the average delay of the proposed delay-constrained protocols are evaluated by analyzing the Markov chain of the states of the queues. Numerical results show that the proposed protocols outperform existing bidirectional relaying protocols for delay-constrained transmission.